Propositional inquisitive logic is the limit of its n-bounded approximations. In the predicate setting, however, this does not hold anymore, as discovered by Ciardelli and Grilletti [11], who also found complete axiomatizations of n-bounded inquisitive logics \(\textsf{InqBQ}_{n}\) , for every fixed n. We introduce cut-free labelled sequent calculi for these logics. We illustrate the intricacies of schematic validity in such systems by showing that the well-known Casari formula is atomically valid in (a weak sublogic of) predicate inquisitive logic \(\textsf{InqBQ}\) , fails to be schematically valid in it, and yet is schematically valid under the finite boundedness assumption. The derivations in our calculi, however, are guaranteed to be schematically valid whenever a single specific rule is not used.